A Topological Picturebook by George K. Francis

By George K. Francis

Goals to inspire mathematicians to demonstrate their paintings and to assist artists comprehend the information expressed by way of such drawings. This booklet explains the photograph layout of illustrations from Thurston's global of low-dimensional geometry and topology. It offers the rules of linear and aerial standpoint from the point of view of projective geometry.

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In designing a line pattern for a surface one is sometimes able to place the border curves with confidence, only to wonder where the contours should go. To obtain a plausible answer I smoothe a piecewise flat and cornered version of the surface. Each line on such a polyhedral swface is an edge bordering one face, or separating two or more faces. Because it is flat, each 18 Figure 3 A TOPOLOGICAL PICTUREBOOK TRIPRONG face of the polyhedron is either wholly visible or wholly invisible. A simple edge separates only two faces, one of which might be hidden.

The algorithm used by Stephen Gray for these graphs of trigonometric polynomials is quite simple and works well though slowly in BASIC on an Apple. Originally, it took ten dedicated minutes in the dead of night on PLATO's Cyber73/CDC6500 timeshared mainframe. It was the convenience of the PLATO system and the uncommonly supportive personal environment that got me started in programming there rather than on a proper graphics setup. My computer novitiate thus was similar to what awaits current beginners in micro-computing.

On the left is an intermediate stage of the disc which I drew to help visualize the former. Serendipity, however, intervened. The helper picture already solves the problem if you use a conjugate of the given relator. This is done in the second picture. The safety-pin serves as the base point. As long as the diaper is pinned, it cannot slip off the knot. This picture was now simple enough to remember and reproduce on demand. The green one in the third photo was drawn in "real time" in Benno Artmann's seminar in Darmstadt.

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